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Full text of "Treatise On Analysis Vol-Ii"

64   XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

(12.14.5)    A locally convex Hausdorff space whose topology can be defined by a
denumerable family of seminorms is metrizable.

This follows from (12.4.6).

A locally convex Hausdorff space E whose topology can be defined by a
denumerable family of seminorms, and which is complete with respect to
every translation-invariant distance on E (12.9.2) is called a Frechet space. A
Banach space (5.1) is therefore a Frechet space.

Example

(12.14.6) Let X be a separable, locally compact, metrizable space and let
^c(X) denote the set of continuous mappings of X into C. Clearly ^CPO is a
vector space over C. Let (Un) be a sequence of relatively compact open sets
in X satisfying the conditions of (3.18.3). For each continuous function
/e ^C(X) and each integer /?, let

(12.14.6.1)                              (/)= sup |/(*)|.

xeUn

It is immediately seen that the/?,, are seminorms on ^C(X) (but they are not
norms if X is not compact). Furthermore, for each /V 0 in #C(X), there
exists a point x e X such that f(x) -fc 0 and an integer n such that x e Un,
hence pn(f) ^ 0. The locally convex space ^C(X) so defined is therefore
metrizable. Moreover, to say that a sequence (fk) in ^C(X) is a Cauchy
sequence signifies (by virtue of (12.9.2)) that, for each integer , the sequence of
restrictions fk \ On is a Cauchy sequence in the Banach space #c(0n) (7.2.1),
hence converges uniformly in On to a continuous mapping gn of Un into C.
Since it is clear that #n+i | t)n =#, there exists a continuous function
/e #CPO whose restriction to each Uw agrees with the restriction of gn.
Evidently lim pn(ffm) = 0 for all n, hence/is the limit of the Cauchy

W1-+ 00

sequence (/m). Thus we have proved that #C(X) is a Frechet space. When X is
compact, it is just the Banach space #C(X) defined in (7.2). Moreover, (7.4.4)
generalizes as follows:

(12.14.6.2)    The Frechet space #C(X) is separable. More precisely, there exists
a denumerable dense set in ^C(X) consisting of continuous functions with com-
pact supports.

With the notation used above, for each integer n there exists a dense
sequence (/zmrt)M^! in the Banach space Vc(0^9 by (7.4.4). By the Tietze-
Urysohn theorem (4.5.1) applied to the closed set Un n (JUn+1 in X, there is a seminorm.r assertion (12.2.1).